Book review for“Fermat’s Last Theorem”
by Takeshi Saito

modifications already present in [1] and [2] to alarge degree. As I am not a specialist in the field
myself, I cannot comment on whether with more
recent methods one could simplify part of theproof. However I am certain that the two books
under review provide a very good base for reading more recent articles in this active researcharea.
Obviously, one should not expect that thecomplete proof is contained in these two volumes. For instance, no reader will be surprised
to see that neither Falting’s proof of the Mordell
conjecture nor the proof of global duality in Galois cohomology is included. The author had tomake some choices as to what prerequisites heassumes and what material he leaves out. Forinstance, despite the rather gentle first 25 pages,
the reader needs to know quite a bit of scheme
theory. Although it is not strictly necessary, it isalso good if the reader is familiar with the basics
theory of elliptic curves and modular forms.
In my opinion, the author makes an excellent
choice as to what to include in order to obtaina coherent exposition which is as self-contained
as possible. The references at the end containprecise indications on where to find the proofs
which are omitted in the books. For instance,
rather than including the full proof of Ribet’s
level lowering theorem, only Mazur’s earlier theorem is given. The reader still gets a good graspof how to obtain such results. The only omission
I felt a bit disappointed about was the lack of discussion on the crucial result by Langlands andTunnell. The Galois representation associated tothe 3-torsion points of an elliptic curve becomesmagically modular with one reference while theremaining part of the book is then about how todeduce from this the modularity of the elliptic
curve itself.
Let me give an overview of the contents ofI was excited to hear that Takeshi Saito’s bookson the proof of Fermat’s Last Theorem had
been translated and appeared as two fairly thinbooks in the series “Translation of Mathematical
Monographs” by the AMS. Richard Taylor andAndrew Wiles made this major breakthrough inthe mid-nineties. Saito’s books, containing boththe “Basic Tools” as well as “The Proof”, appear
twenty years later with these subtitles.
The subject has lost none of its appeal andimportance in the interim; on the contrary, themethods used in the proof produced numerous
other important results, such as the proof of theSato-Tate conjecture by Richard Taylor and his
collaborators and the proof of Serre’s modularity
conjecture by Chandrashekhar Khare and JeanPierre Wintenberger.
There have been quite a number of publications devoted to the mathematics involved in
the the proof of Fermat’s Last Theorem. Mostof them are surveys or they present the mainmathematical concepts (elliptic curves, modular forms and Galois representations) involved in
the proof without going into details. The conference volume [1] edited by Cornell, Silverman andStevens and the article [2] by Darmon, Diamondand Taylor stand out as having a similar aim as
the books under review. The former is a collection of very good articles, though it may lack in
consistency and clear focus which is one of thestrengths of Saito’s books. The latter does notcover the basic tools in as much detail and itis less self-contained. All three are best used inparallel.
The books follow the original arguments and1

ramification only outside a finite set Σ of primes.
The author defines what he calls an RTM-triple,which consists of triples (RΣ , TΣ , MΣ ) together
with a map f : RΣ → TΣ . Two numerical criteria
for a ring like RΣ to be a local complete intersection and for the map f to be an isomorphism
are stated. The first will be used to prove thecase when Σ is empty and the second to prove it
inductively when enlarging the set Σ.
The first book finishes with two more technicalchapters, one on commutative algebra proving
the two criteria for local complete intersections
and the other on deformation rings of Galois representations. In the latter the construction of theuniversal deformation ring RΣ is given.
The second book starts with chapter 8, whichcontains a thorough and nice development ofmoduli problems for elliptic curves with the aimof constructing the modular curves as schemes
over Z. The curves Y0 (N ) and Y1 (N ), theircompactifications and the important maps between them are carefully constructed. More exotic moduli problems and the Igusa curves arealso explained. They are used in the next chapter which starts by constructing the Hecke algebra with integer coefficients and the Galois representation attached to a modular form. The
second part of this chapter is devoted to the level
lowering result studying the Néron model of theJacobian of the modular curve.
The heart of the proof that RΣ and TΣ areisomorphic is contained in the last two chapters. First, in chapter 10, the Hecke module MΣ
coming from modular symbols is defined andanalysed. The maps between modular curves
are used to compute a so-called multiplier andshow the surjectivity of a map between the Hecke
modules as Σ increases. Then one has to verify
the numerical criterion using a well-chosen set Qof auxiliary primes. The final chapter introduces

the eleven chapters of the books. The first bookstarts with a synopsis containing the statementof Fermat’s Last Theorem and an explanation of
how the modularity of the Frey curve implies it.
Then the first chapter covers elliptic curves
over arbitrary base schemes and generalised elliptic curves in view of the moduli problems described later. As mentioned before, this is selfcontained but having seen a more elementary
approach before may be helpful. The following chapter covers the basics on modular forms
of weight 2. They are systematically developed
as geometric modular forms by constructing themodular curves as moduli spaces over Q andmodular forms are defined as differentials on
them. Chapter 3 introduces Galois representations and local conditions on them. It is herethat we meet for the first time the crucial condition simply called “good at p” which requires
the local representation to come from a finite flatgroup scheme defined over Zp . It is explained
when this is the case for the Galois representation attached to an elliptic curve and stated for
the Galois representation attached to a modularform.
After having set up the three main concepts
that are involved in the proof, the next two chapters give an overview of the proof and announce
the major theorems that will be proven later. In
chapter 4, the 3–5 trick explains how the modular lifting theorem implies the modularity ofsemi-stable elliptic curves. An overview of theproof of the modular lifting theorem is given in
the next chapter. It explains the meaning anduse of what is simply called R = T , namely thata certain universal deformation ring of Galois
representations is isomorphic to a certain Hecke
algebra connected to modular forms. Both ofthem are described together with finer versionsRΣ and TΣ which impose the local restrictions on2

Galois cohomology. The relevant Selmer groups References
are defined and linked to the deformation problem of Galois representations. Finally, global du- [1] Gary Cornell, Joseph H. Silverman, andGlenn Stevens, editors. Modular forms andality of Galois cohomology concludes the proof.
Fermat’s last theorem. Springer-Verlag, NewThe books also have appendices with collecYork, 1997. Papers from the Instructionaltions of results on arithmetic geometry, on theConference on Number Theory and Ariththeory of Fontaine and Laffaille and on Néronmetic Geometry held at Boston University,models. They have good indices of notations, aBoston, MA, August 9–18, 1995.
complete bibliography, but they contain hardly[2] Henri Darmon, Fred Diamond, and Richard
any examples or exercises.
Taylor. Fermat’s last theorem. In Ellipticcurves, modular forms & Fermat’s last theAs I reach the conclusion of this review, Iorem (Hong Kong, 1993), pages 2–140. Int.
would like to mention one issue I had with thesePress, Cambridge, MA, 1997.
books: The logical structure is very complicated.
Frequently, theorems are announced and only
proven much later after they were used to deduce Christian Wuthrichearlier statements. It is a matter of taste, but University of Nottingham
I would have preferred a more linear structureto these frequent flash-forwards. Sections 4.2,
5.5 and 5.6 are of much help with understanding
the structure.
Another strange, yet harmless fact about thebook is that it omits to attribute any of the intermediate results to the mathematicians who
discovered them. Also famous buzzwords like“modularity lifting” or “level lowering” are notmentioned.
In summary, I enjoyed reading this book. The
author states in his introduction “I would be extremely gratified if more people could appreciate
one of the highest achievements of the twentieth century in mathematics.” These books are avery good contribution for exactly this purpose.
Number theorists, including graduate students,
will find the proof of Fermat’s Last Theorem andthe modularity of elliptic curves and its prerequisites much more accessible thanks to these twobooks.
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